2-local real-linear isometries on C ⁽¹⁾([0, 1])

Abstract

Let C ⁽¹⁾([0, 1]) be the Banach space of continuously differentiable functions on the closed unit interval [0, 1] equipped with the norm ||f||_σ= | f (0)| +|| f ′||_∞, where ||g||_∞= sup{|g(t)| : t ∈ [0, 1]} for g. If T : C ⁽¹⁾ ([0, 1]) → C ⁽¹⁾ ([0, 1]) is a 2-local real-linear isometry, then T is a surjective real-linear isometry.